Delta sigma (ΔΣ) modulators are known in the prior art and may be used in connection with analog-to-digital (A/D) conversion of data (signals) for high-IF subsampling. Typical high frequency A/D converters have high frequency sample and hold circuits. Unfortunately, as the A/D conversion frequency increases, the sample and hold circuits become more and more imprecise. Continuous-time ΔΣ A/D converters do not need sample and hold circuits at the input. But while continuous-time ΔΣ A/D converters offer high-IF resolution, they have a low precision sampler (which can be as imprecise as 1 bit).
A large amount of literature has been devoted to continuous-time ΔΣ modulators (CT-ΔΣM's) that use OTA-C (operational transconductance amplifier capacitor) or LC (inductor capacitor) resonators. See, for example, (1) G. Raghavan, et al., “Architecture, design, and test of continuous-time tunable intermediate-frequency bandpass delta-sigma modulators,” IEEE Journal of Solid-State Circuits, vol. 36, no. 1, January 2001, pp. 5–13 and (2) J. van Engelen et al., Bandpass Sigma Delta Modulators: Stability Analysis, Performance and Design Aspects, Kluwer Academic Publishers, 1999. The dynamic range of a CT-ΔΣM can improve by a factor of Q02 over the dynamic range of an OTA-C resonator, where Q0 is the unloaded Q of the resonator. See W. B. Kuhn, F. W. Stephenson and A. Elshabini-Riad, “Dynamic Range of high-Q OTA-C and enhanced-Q LC RF bandpass filters,” Proceedings of the 37th Midwest Symposium on Circuits and Systems, 1994, vol. 2, pp. 767–771. While LC resonators typically have Q0's around 10, distributed resonators can have Q0's in the hundreds.
To realize a system with a distributed resonator, one would, for example, input current into the resonator and measure its voltage. For the rest of this paper, we will assume that we are using a transmission-line resonator where one forces a current into the resonator and measures the voltage, but the proposed method will also work for arbitrary distributed resonators, and additionally the proposed method will work for designs where one forces a voltage across the resonator and measures the current. The impedance of a transmission-line resonator would then form the transfer function.
FIG. 1 depicts a grounded transmission-line resonator and an open transmission-line resonator. Examining FIG. 1, let δT be the propagation delay through the resonator and T be the sampling time. Once the substitution βl=ωδT is made, then the impedance is given as (see D. Pozar, Microwave Engineering, Addison-Wessley, Massachusetts, 1990, pp. 336–343):
                                          Z            in                    ⁡                      (            s            )                          =                  {                                                                                                                jZ                      0                                        ⁢                                          tan                      ⁡                                              (                                                  ωδ                          ⁢                                                                                                          ⁢                          T                                                )                                                                              =                                                            Z                      0                                        ⁢                                                                  1                        -                                                  ⅇ                                                                                    -                              2                                                        ⁢                            s                            ⁢                                                                                                                  ⁢                            δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                      1                        +                                                  ⅇ                                                                                    -                              2                                                        ⁢                            s                            ⁢                                                                                                                  ⁢                            δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                ⁢                                          (                      grounded                      )                                                                                                                                                                                      jZ                      0                                        ⁢                                          cot                      ⁡                                              (                                                  ωδ                          ⁢                                                                                                          ⁢                          T                                                )                                                                              =                                                            Z                      0                                        ⁢                                                                  1                        +                                                  ⅇ                                                                                    -                              2                                                        ⁢                            s                            ⁢                                                                                                                  ⁢                            δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                      1                        -                                                  ⅇ                                                                                    -                              2                                                        ⁢                            s                            ⁢                                                                                                                  ⁢                            δ                            ⁢                                                                                                                  ⁢                            T                                                                                                                ⁢                                          (                      open                      )                                                                                                                              (        1        )            
Reference will be made to the case of the grounded resonator, although the following analysis can easily be generalized to the open resonator. If the resonator's fundamental resonance frequency is f0, then there are additional resonances at 3f0, 5f0, 7f0 and so on. ΔΣ modulators comprise a feedback DAC. If the feedback DAC generates output components at these frequencies, the harmonics will resonate and influence the signal into the quantizer. At best, these harmonics create significant aliases. Let FS=the sampling frequency and let fn=the harmonic frequency. We let kn=mod(fn,FS). Then, the alias frequencies after sampling are given as:
                              a          n                =                  {                                                                                          k                    n                                    ,                                                                                                  k                    n                                    <                                                            F                      S                                        /                    2                                                                                                                                                                                          F                        S                                            /                      2                                        -                                          k                      n                                                        ,                                                                                                  k                    n                                    ≥                                                            F                      S                                        /                    2                                                                                                          (        2        )            
These aliases can severely degrade the SNR or cause the modulator to go unstable.
A possible method to overcome this problem is that of decreasing the high-frequency DAC components by slowing the DAC down. However, in this way the likelihood of intersymbol interference is increased, because the DAC may not fully settle by the time it needs to switch again.
The present disclosure offers an alternative method to overcome the above shortcomings.